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From Backward Approximations to Lagrange Polynomials in Discrete Advection–Reaction Operators

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Abstract

In this work we introduce a family of operators called discrete advection–reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection–reaction–diffusion partial differential equations. They consists of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.

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Notes

  1. A polynomial with all its coefficients equal to one.

References

  1. Asheghi, R.: Bifurcations and dynamics of a discrete predator-prey system. J. Biol. Dyn. 8(1), 161–186 (2014)

    Article  MathSciNet  Google Scholar 

  2. Bischi, G.I., Naimzada A.K.: A Kaleckian macromodel with memory. In: Cristini, A., Fazzari, S., Greenberg, L., Leoni R. (eds) Cycles, Growth and the Great Recession, Routledge, United Kingdom, pp 103–116 (2015)

  3. Burden, R.L.: Numerical analysis. Beooks/Cole, Australia (2001)

    Google Scholar 

  4. Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Sohaly, M.A.: Solving the random Cauchy one-dimensional advection-diffusion equation: numerical analysis and computing. J. Comput. Appl. Math. 330, 920–936 (2018). https://doi.org/10.1016/j.cam.2017.02.001

    Article  MathSciNet  MATH  Google Scholar 

  5. Cortés, J.C., Jódar, L., Villafuerte, L., Villanueva, R.J.: Computing mean square approximations of random diffusion models with source term. Math. Comput. Simul. 76(1–3), 44–48 (2007). https://doi.org/10.1016/j.matcom.2007.01.020. 12

    Article  MathSciNet  MATH  Google Scholar 

  6. Chong, T.H.: A variable mesh finite difference method for solving a class of parabolic differential equations in one space variable. SIAM J. Numer. Anal. 15, 835–857 (1978)

    Article  MathSciNet  Google Scholar 

  7. El-Sayed, A.M.A., Elsadany, A.A., Awad, A.M.: Chaotic dynamics and synchronization of cournot duopoly game with a logarithmic demand function. Appl. Math. 9(6), 3083–3094 (2015)

    Google Scholar 

  8. Forsythe, G.E., Wasow, W.R.: Finite difference methods for partial differential equations. Wiley, New York (1964)

    MATH  Google Scholar 

  9. Gladwell, I., Wait, R.: A survey of numerical methods for partial differential equations. Oxford University Press, New York (1979)

    MATH  Google Scholar 

  10. Jerez, S., Gonzalez, L.M., Solis, F.J.: A regular perturbation analytical-numerical method for the evolution of precancerous lesions caused by the human papillomavirus. Numer. Methods Partial Differ. Eq. 31(3), 847–855 (2015)

    Article  MathSciNet  Google Scholar 

  11. Khodabin, M., Maleknejad, K., Rostami, M., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge-Kutta methods. Math. Comput. Model. 53, 1910–1920 (2011)

    Article  MathSciNet  Google Scholar 

  12. Li, B., He, Z.: Bifurcations and chaos in a two-dimensional discrete HindmarshRose model. Nonlinear Dyn. 76(1), 697–715 (2014)

    Article  Google Scholar 

  13. Maire, S., Nguyen, G.: Stochastic finite differences for elliptic diffusion equations in stratified domains. Math. Comput. Simul. 121, 146–165 (2016)

    Article  MathSciNet  Google Scholar 

  14. Pareek, N.K., Patidar, V., Sud, K.K.: Discrete chaotic cryptography using external key. Phys. Lett. A 309, 75–82 (2003). https://doi.org/10.1016/S0375-9601(03)00122-1. 12, 17

    Article  MathSciNet  MATH  Google Scholar 

  15. Reynolds Jr., A.C.: Convergent finite difference schemes for non-linear parabolic equations. SIAM J. Numer. Anal. 9, 523–533 (1972)

    Article  MathSciNet  Google Scholar 

  16. Schulman, L.S., A Seiden, P.E.: Statistical mechanics of a dynamical system based on Conways game of Life. J. Stat. Phys. 19(3), 293–314 (1978)

    Article  MathSciNet  Google Scholar 

  17. Solis, F., Gonzalez, L.M.: A model for HVP infected cells at different lesion discrete stages. Int. J. Complex Syst. Sci. 2(1), 7–10 (2012)

    Google Scholar 

  18. Solis, F., Gonzalez, L.M.: Modeling the effects of Human Papilloma Virus in cervical cells. Int. J. Comput. Math. 91, 1–9 (2013)

    Google Scholar 

  19. Solis, F., Barradas, I.: Discrete multiple delay advection-reaction operators. J. Comput. Appl. Math. 291, 441–448 (2016)

    Article  MathSciNet  Google Scholar 

  20. Solis, F.: Dynamical properties of families of discrete delay advection-reaction operators. J. Differ. Equ. Appl. 19(8), 1218–1226 (2013)

    Article  MathSciNet  Google Scholar 

  21. Solis, F.: Families of discrete advection-reaction operators via divided differences. Appl. Math. Lett. 25, 775–778 (2012)

    Article  MathSciNet  Google Scholar 

  22. Vasilyev, R.V., et al.: Solution of the stokes equation in three-dimensional geometry by the finite-difference method. Math. Models Comput. Simul. 8(1), 63–72 (2016)

    Article  MathSciNet  Google Scholar 

  23. Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25(2), 351–375 (1988)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CIMAT, Apartado Postal 402, 36000, Guanajuato Gto, Mexico

    Francisco J. Solis, Ignacio Barradas & Daniel Juarez

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  1. Francisco J. Solis
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  2. Ignacio Barradas
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  3. Daniel Juarez
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Correspondence to Francisco J. Solis.

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Solis, F.J., Barradas, I. & Juarez, D. From Backward Approximations to Lagrange Polynomials in Discrete Advection–Reaction Operators. Differ Equ Dyn Syst 29, 363–375 (2021). https://doi.org/10.1007/s12591-018-0415-9

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